In your question you mentioned the word "Fredholm index".

So I would like to say that in the circle case there are two different interpretations of Fredholm index of certain linear operators in terms of winding number. So it would be interesting to consider a possible generalization of these $1$ dimensional facts to higher dimensional spheres.

1)If I remember the following theorem correctly, there is a Theorem by Veku in "Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics, by B. Booss and D. D. Bleecker" which says:

**Theorem:** If $X$ is a non vanishing vector field along $S^1\subset \mathbb{R}^2$, not necessarily tangent to $S^1$, then the fredholm index of the pair operator $(\Delta, \partial/\partial X)$ is equal to the "winding number of $X$. In the above pair operator $\Delta$ is the standard Laplace operator on the interior of circle and the derivational operator $\partial/\partial X$ is restricted to the boundary.

So it would be a good idea to consider an appropriate generalization of this fact. For every smooth self map on sphere one can consider an arbitrary extension to whole $\mathbb{R}^{n+1}$ and try to find an appropriate generalization. However an immediate plain generalization is not true but one should consider a modified generalization. I was thinking to this question about 7 years ago and I observed that a plain generalization is not true since the corresponding operator on $S^3$ is not a Fredholm operator. I had intension to discuss these materials in my following talk I presented in Timisoara but I did not had enough time to present all of the materials of this abstract, since my time was 20 minutes:

http://at.yorku.ca/c/b/d/z/37.htm

(Sorry, the talk abstract I wrote is very snafu)

2) Many years ago I learned from a speaker in Non commutative geometry

that the multiplication by a non vanishing complex map $f$ on $S^1$ is a fredholm operator on $L^1(S^1)$ whose index is $- W(f)$, the winding number of $f$. So it would be interesting to consider an $S^3$ analogy or even more a compact Lie group analogy.

The following paper contains an explanation of a similar situation

http://www.ams.org/journals/proc/2005-133-05/S0002-9939-04-07642-7/S0002-9939-04-07642-7.pdf

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